Context: Most globular clusters are believed to host a single stellar population. They can thus be considered a good place to study the Spite plateau and to search for possible evolutionary modifications of the Li content. Aims: We want to determine the Li content of subgiant (SG) and main sequence (MS) stars of the old, metal-poor globular cluster NGC 6397. This work was aimed not only at studying possible Li abundance variations but also to investigate the cosmological Li discrepancy. Methods: Here, we present FLAMES/GIRAFFE observations of a sample of 84 SG and 79 MS stars in NGC 6397 selected in a narrow range of B-V colour and, therefore, effective temperatures. We determine both effective temperatures and Li abundances using three-dimensional hydrodynamical model atmospheres for all the MS and SG stars of the sample. Results: We find a significant difference in the Li abundance between SG stars and MS stars, the SG stars having an abundance higher by almost 0.1 dex on average. We also find a decrease in the lithium abundance with decreasing effective temperature, both in MS and SG stars, albeit with a significantly different slope for the two classes of stars. This suggests that the lithium abundance in these stars is, indeed, altered by some process, which is temperature-dependent. Conclusions: The lithium abundance pattern observed in NGC 6397 is different from what is found among field stars, casting some doubt on the use of globular cluster stars as representative of Population II with respect to the lithium abundance. None of the available theories of Li depletion appears to satisfactorily describe our observations. Based on observations obtained with FLAMES/GIRAFFE at VLT Kueyen 8.2 m telescope in programme 079.D-0399(A). Table and Figs. 3-10 are only available in electronic form at http://www.aanda.org Table 2 is available in electronic form at http://www.aanda.org and at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/505/L13